Mixing of the symmetric beta-binomial splitting process on arbitrary graphs

We study the mixing time of the symmetric beta-binomial splitting process on finite weighted connected graphs $G=(V,E,\{r_e\}_{e\in E})$ with vertex set $V$, edge set $E$ and positive edge-weights $r_e>0$ for $e\in E$. This is an interacting particle system with a fixed number of particles that updates through vertex-pairwise interactions which redistribute particles. We show that the mixing time of this process can be upper-bounded in terms of the maximal expected meeting time of two independent random walks on $G$. Our techniques involve using a process similar to the chameleon process invented by Morris (2006) to bound the mixing time of the exclusion process.